The response of two-degree-of-freedom systems with quadratic non-linearities to a combination parametric resonance |
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| Affiliation: | 1. Institute for Advanced Materials and Technology, University of Science and Technology Beijing, Beijing 100083, China;2. Department of Materials Science and Engineering, Cornell University, Ithaca, New York 14853, United States;3. Beijing Key Laboratory for Advanced Powder Metallurgy and Particulate Materials, University of Science and Technology Beijing, Beijing 100083, China;4. School of Material Science and Engineering, Xiangtan University, Hunan 411105, China;5. Beijing Advanced Innovation Center for Materials Genome Engineering, University of Science and Technology Beijing, Beijing 100083, China;1. Department of Electronic Science and Engineering, Kyoto University, Katsura, Nishikyo, Kyoto 615-8510, Japan;2. Department of Electrical Engineering, Kyoto University, Katsura, Nishikyo, Kyoto 615-8510, Japan;1. State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, China;2. Dalian Municipal Design and Research Institute of Environmental Science, Dalian 116024, China |
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| Abstract: | The response of two-degree-of-freedom systems with quadratic non-linearities to a combination parametric resonance in the presence of two-to-one internal resonances is investigated. The method of multiple scales is used to construct a first order uniform expansion yielding four first order non-linear ordinary differential equations governing the modulation of the amplitudes and the phases of the two modes. Steady state responses and their stability are computed for selected values of the system parameters. The effects of detuning the internal resonance, detuning the parametric resonance, the phase and magnitude of the second mode parametric excitation, and the initial conditions are investigated. The first order perturbation solution predicts qualitatively the trivial and non-trivial stable steady state solutions and illustrates both the quenching and saturation phenomena. In addition to the steady state solutions, other periodic solutions are predicted by the perturbation amplitude and phase modulation equations. These equations predict a transition from constant steady state non-trivial responses to limit cycle responses (Hopf bifurcation). Some limit cycles are also shown to experience period doubling bifurcations. The perturbation solutions are verified by numerically integrating the governing differential equations. |
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